The semi-simple zeta function of quaternionic Shimura varieties
This monograph is concerned with the Shimura variety attached to a quaternion algebra over a totally real number field. For any place of good (or moderately bad) reduction, the corresponding (semi-simple) local zeta function is expressed in terms of (semi
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Harry Reimann
The semi-simple zeta function of quaternionic Shimura varieties
Springer
Author Harry Reimann Fakultat fur Mathematik Universitat Bielefeld PF 100131 D-33501 Bielefeld, Germany e-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Reimann, Harry: The semi-simple zeta function of quaternionic Shimura varieties / Harry Reimann. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong ; London; Milan ; Paris ; Santa Clara; Singapore; Tokyo: Springer, 1997 (Lecture notes in mathematics ; 1657) ISBN 3-540-62645-X NE:GT
Mathematics Subject Classification (1991): I1G18, 11G40, 14G10, 14035, 11F70, IlF72, 11G15, 14KlO ISSN 0075-8434 ISBN 3-540-62645-X Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10520361 46/3142-543210- Printed on acid-free paper
Fur Gabi, Kevin, Sarah und Tobias
Preface The aim of this monograph is the study of the Shimura variety ShD,c attached to the multiplicative group of a quaternion division algebra D over a totally real number field F and a sufficiently small open compact subgroup C of D x (Ai)' For every rational prime p which is unramified in F and such that C = Cp x CP where Cp is a maximal compact subgroup of D X (lQlp) and CP is a subgroup of DX(Aj), the semisimple local zeta function is expressed as a product of integral powers of semisimple local Lfunctions of automorphic representations of D X (A). In an appendix a mistake occurring in the constructions of Langlands and Rapoport needed to state their conjecture on the reduction of a Shimura variety in the most general case is corrected, and the conjecture is restated in a slightly stronger form. It is proved for ShD,c up to a question concerning the reduction of abelian varieties of eM type which is discussed in another appendix. If, however, either the quaternion algebra D is totally indefinite or the condition for the presence of good reduction whi
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